Oberseminar Algebra und Geometrie

Short description

The aim of the seminar is to get to know arithmetic groups. We will mainly follow the self-contained Lecture Notes [Hu80] of James E. Humphreys. In particular, we cover: the necessary number theoretic background in the setting of locally compact abelian groups and discrete subgroups; the general linear and special linear groups, with an emphasis on “reduction theory” (computation of fundamental domains; information about finite generation and finite presentations); the Congruence Subgroup Problem. A few other sources, including the books [Mo15] by Morris Witte and [PR94] by Platonov and Rapinchuk, are listed below. Due to time considerations, we will not prove all, but only selected results. In particular, the last two talks are meant to survey technical results rather than to treat them in detail.

Schedule

13.04.18

1. Locally compact groups and fields (Florian Severin)

Main source: Ch. I, Sec 1-3 of [Hu80]

Keywords: Haar measure, module of an automorphism, local and global fields, classification and strucutre theorems, adele ring of a global field.

20.04.18

2. The additive group (Kevin Langlois)

Main source: Ch. II, Sec 4-6 of [Hu80]

Keywords: The quotient of the adele group by the number field, fundamental domains and product formula, volume of a fundamental domain, Strong Approximation Theorem.

27.04.18

3. The multiplicative group & Example: The modular group (Johannes Fischer)